When Prometheus handed Homo sapiens the gift of fire, he did not just give humanity the light of science, reason, progress, invention, technology, and ergo the power to rival God, but also the means to dispatch just about any monster imaginable. Considering how Greek Mythology is a Fantasy Kitchen Sink full of all kinds of abominations, that was a good thing. The symbolism behind this has to do with fire's associations with purification and light, and partly because it represents humanity's dominion over the natural world. More literally, a burning stick was humankind's first effective defence against nocturnal predators, with an added bonus of extermination of parasites in food. The fact that burn pain is one of the strongest kinds of pain a human being can experience probably doesn't hurt either.
Also, in a pinch, it can still burn things, or at least scare them off. Arrows on Fire, Flaming Swords, Molotov Cocktails, flamethrowers, or good old Torches and Pitchforks can work wonders when dealing with everything from Frankenstein's Monster to a Monster House.
This is prevalent not just in myth and fiction or games based on it, but also in works that are completely new and unrelated. For example, the Hydra and Trolls demonstrate one of the most frequent, and logical, applications for anti-monster fire: It prevents regeneration. In the case of The Undead, usually they're too dumb and slow to put it out (zombies), it reminds them of the sun (vampires), or they're already walking kindling (mummies). Other times it's used to make sure whatever just got killed stays that way. Aliens, of course, can be Immune to Bullets, but will burn up nicely in a fire. Witches and heretics for a long time were considered to be in league with evil as well, and so it was a common myth that they were dealt with in the same manner. And, needless to say, it's a very good strategy for When Trees Attack.
You know what the best part of Kill It with Fire is? It works on regular people wearing masks too. Kind of like decapitation (which also works on most things) and a Wooden Stake through the heart.
If something's too big or tough for a regular fire, you can always use plasma or Hurl It into the Sun. This has also been subject to Memetic Mutation, as an alternative name for Brain Bleach.
A character who's a little too enthusiastic about using this can become a Pyromaniac. A character may try in vain to use this trope, but be faced with a Hard-to-Light Fire.
A Super-Trope to Matchstick Weapon and Fire Keeps It Dead. See also Incendiary Exponent, Man on Fire, Burn Baby Burn, Fire-Breathing Weapon, Burn the Undead, Homemade Flamethrower, Burn the Witch!, Fanatical Fire, Gasoline Dousing, Weak to Fire and Hellfire (for when mundane fire doesn't do the job). Compare Stuff Blowing Up and Playing with Fire — for when someone has fire as a superpower, and Murder by Cremation — when an ordinary person is lit on fire. When this just makes things worse, see Infernal Retaliation. May also overlap with A Molten Date with Death if lava is involved. The less-used opposite is either Kill It with Water or Kill It with Ice, or even Heal It With Fire. Contrast Immune to Fire.
If you were looking for the video game by the same name, you'll find it here.
NOTE: Due to the above-mentioned Memetic Mutation, the name of this trope has become associated with anything that someone finds abhorrent and wishes to not have witnessed. This trope is not about those things. Please redirect all potholes using this trope in said context to Brain Bleach or a similarly appropriate trope.